2742
Macromolecules 1990, 23, 2742-2748
However, in our previous work we have found that in PMMA the E(CQ)/E(CQP) ratio is equal to 1.03, and in p~lycarbonate’~ it is equal to 1.34. These results thus demonstrate that the { value and hence for the critical molecular volume V d * of the dye dependents on the nature of the polymer host. Using eq 6, we find that above Tgthe effective activation energy
E, = R T ~ ( % )
=RT+
Acknowledgment. This work is financially supported by ONR and by the Polymer Program (DMR8606884) of the National Science Foundation. References a n d Notes (1) Zhang, J.; Wang, C. H. Macromolecules 1988, 21, 1811 and
references cited herein. (2) Xia, J. L.; Wang, C. H. J . Chem. Phys. 1988, 88, 5211. (3) Ehlich, D.; Sillescu, H. Tracer diffusion at the glass transi-
tion. a reprint. (4) Searle, 0. B.; Pfeiffer, R. H. Polym. Eng. Sci. 1985, 25 (81,
P
2.3[C1C&T
(C,+T-TJ Equation 10 indicates that the effective activation energy increases rapidly as T is decreased toward T,. Furthermore, one notes that ED is linearly proportional to {. Thus, given the same polymer environment, the rate of dye diffusion in a polymer is strongly affected by the size of the dye molecule. In conclusion, we have carried out diffusion measurements of a trace amount of CQ in PEEK as a function of temperature. We have shown that the diffusion coefficient of CQ in PEEK is closely associated with the viscoelastic property of PEEK. The main-chain segmental motion that affects the dynamic shear modulus data also dominates the diffusion process of camphorquinone and its photoproduct in PEEK.
474. Sasuga, T.; Hagiwara, M. Polymer 1985,26, 501. Zhang, H. C.; Chen, T. L. Chinese Patent 85108751, 1985. Monroe, B. M. Adu. Photochem. 1971,8,77. Wang, C. H.; Xia, J. L. J . Chem. Phys., in press. Li, B. Y.; Zhang, H. C.; Yang, J. M.; Ni, S. R.; Chen, T. L. Polym. Mat. Sci. Eng. 1987, 3, 35. (10) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (11) Fujita, H. Fortschr. Hochpo1ym.-Forsch.1961, 3, 1. (12) Vrentas, J. S.; Liu, H. T.; Duda, J. L. J . Appl. Polym. Sci. 1980,25, 1297. Vrentas, J. S.; Duda, J. L.; Ling, H. C . J . Polym. Sei., ‘Polym.Phys. Ed. 1985, 23, 275. Vrentas, J. S.; Duda, J. L.; Ling, H.-C.; Hou, A. C. J . Polym. Sci., Polym. Phys. Ed. 1985,23,289. Vrentas, J. S.;Duda, J. L.; Hou, A. C. J. PoEym. Sci., Polym. Phys. Ed. 1985, 23, 2469. (13) Zhang, J.; Wang, C. H. J . Phys. Chem. 1986,90, 2296. Xia, J. L.; Wang, C. H. Mass diffusion of camphorquinone in poly(methyl methacrylate) hosts with different molecular weights. J . Polym. Sci. Phys., in press. (14) Wang, C. H.; Xia, J. L. Macromolecules 1988, 21, 3519. Registry No. CQ, 465-29-2; PEEK, 40883-84-9. (5) (6) (7) (8) (9)
Chain Architecture in the Hydrodynamic Scaling Picture for Polymer Dynamicst George D.J. Phillies Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 Received September 6, 2989; Revised Manuscript Received November 18, 1989
ABSTRACT: The hydrodynamic scaling picture of polymer dynamics is extended to treat the consequences of chain architecture on tracer diffusion by linear and star polymers in a matrix of linear polymers. From hydrodynamic scaling, one clearly predicts: (i) whereas in comparing linear and many-armed star polymers of equal total molecular weight, a given large concentration of matrix polymer is modestly more effective (smaller DIDO)at retarding a linear polymer than a t retarding a star polymer; (ii) with linear and star polymers of equal arm molecular weight, a large concentration of matrix polymer is far more effective a t retarding a star than a t retarding a linear polymer. Contrary to the claims of their authors, recent experiments of Lodge, Wheeler, et al. (Macromolecules 1989,23, 3409) on tracer diffusion of linear and star polystyrenes in poly(viny1 methyl ether) matrices are consistent with predictions of the hydrodynamic scaling model.
Introduction Wheeler, Lodge, and Markland1s2have recently reported an extensive experimental study on the tracer diffusion coefficient D of linear and star polystyrenes (PSI through poly(viny1 methyl ether) (PVME)/o-fluorotoluene, comparison being made with the reptation-~caling,~ reptation+ The support of the donors of the Petroleum Research Fund, administeredby the American Chemical Society, is gratefully acknowledged.
0024-9297/90/2223-2742$02.50/0
M ~ r i ,dynamic ~,~ entanglement,6v7 and hydrodynamic scalings-10 models of polymer dynamics. Lodge et ale2 concluded “...star-branched polymers are retarded significantly relative to linear polymers ...the observation that branched polymers move less rapidly is consistent only with the reptation-based picture.” A reader might reasonably conclude from the latter phrase that Lodge et al.2 viewed their findings to be inconsistent with the hydrodynamic scalings-10 and dynamic entanglement6~7models. Since neither of these models had hitherto been C 1990 American Chemical Society
Hydrodynamic Scaling Picture of Polymer Dynamics 2743
Macromolecules, Vol. 23, No. 10, 1990 extended to treat influences of chain architecture on D , it was clearly erroneous to assert that any finding on chain versus star diffusion could be inconsistent with either hydrodynamic scaling or with dynamic entanglement, at least as those models existed when refs 1 and 2 were written. The objective of this note is to report the incorporation of chain architecture effects into the hydrodynamic scaling model. It will be shown that Lodge, Wheeler, et al.’s1s2experiments on D of linear (0,) and star (D.) polymers are in reasonable agreement with hydrodynamic scaling. A t the present level of sophistication of theory and experiment, comparison of DI and D I does not distinguish between hydrodynamic scaling, dynamic entanglement, and some reptation-based pictures of chain dynamics. The primary experimental data are studies of tracer diffusion by dilute (“probe”)polymer chains through solutions containing a background (“matrix”) polymer; the matrix polymer may be concentrated. If the probe and matrix polymers are chemically identical, tracer diffusion becomes self-diffusion. Experimental findings on tracer diffusion in polymer solutions have recently been reviewed.” Polymer melts are an unrelated issue not considered treated here. Wheeler et a1.lP2 report a detailed study of tracer diffusion, varying in particular the architecture of their probe chains. Their major experimental findings were as follows: (i) D1 and De of tracer polystyrenes fall with increasing concentration c of the PVME matrix polymer, the concentration dependence following the stretched-exponential (so-called1 “Phillies”) form
D = Do exp(-ac”)
(1)
with cy and u as scaling parameters, c being the matrix concentration. Do,if treated as a free parameter, is somewhat larger than the low-c limit of D. a and v depend on probe M and on chain architecture. (ii) Linear and star PS species have different concentration dependences. Comparing polymers of the same total molecular weight M , a t large matrix c many-armed star polymers are less retarded (larger DIDO)by the matrix than are linear polymers. For 12-arm stars and linear (2-arm star) polymers of equal M , [D*/DO,*]/[D~/DO,II reaches a largest value of 2 for c 100 g/L. Above 100 g/L, [D./Do,*]/[DI/Do,~] falls with increasing c. On the other hand, with polymers having equal arm molecular weight Ma (a linear polymer being a 2-arm star), [D*/Do,*]/ [D1/Do,,]falls rapidly with increasing c. A t large matrix concentrations, the retardation of a star polymer by a matrix may be an order-of-magnitude larger than the retardation of a linear polymer having the same Ma. (iii) R, of linear and star polymers, as obtained from static light scattering, falls substantially with increasing
-
n
L.
These experimental data may be compared with four major families of models of polymer solution dynamics. In the reptation models, a concentrated polymer solution is described as a meshwork (“pseudogel lattice”), in which polymer chains have cross-link (“entanglement”) points which constrain polymer chains to diffuse parallel to their own contours. Reptation models may be divided between reptation-scaling model^,^ in which variables are assumed to be related by power-law (“scaling”) relations, and nonscaling models, in which reptation is supplemented by dynamic equations such as the eq 4 of the Mori-Zwanzig projection formalism. The reptation models are here collectively described as “static entangle-
Table I Dependence of the Dilute-Solution Doof 2-, 3-, and 12-Arm Star Polymers on Polymer Molecular Weight M,As Described by Eq 2. 2 3 12
0.538 0.552 0.551
2320 2717 3078
2.4 -0 5.4
a A 2-arm star is a linear polymer; M is in Da. brmse = rootmean-square error.
ment” models because in these models the entanglement points appear and disappear but do not move relative to the solution volume. A third family of models, such as Einaga’s “noodle” picture12 and Skolnick et al.’s6p7 phenomenological picture, resembles the reptation models in that chain entanglements dominate interpolymer interactions. These models differ from the reptation models in that entanglements are dynamic and move through solution on significant time scales. In these “dynamic entanglement” models, polymer chains are constrained by entanglements. However, dynamic entanglements move, so chains can diffuse both parallel and perpendicular to their own contours. Dynamic entanglement models have received extensive support from elaborate on-lattice computer Monte Carlo dynamics calculation^.^^^ A fourth model of polymer solution dynamics is this author’s hydrodynamic scaling picture,8-1° in which the dominant force between polymer chains is assumed to be hydrodynamic in nature, moderated by a Newtonian solvent filling the spaces between the polymer chains. A self-similarity approximation for multiple-chain interactions is used to bootstrap the computation from dilute solution to large concentrations, yielding stretchedexponential concentration dependences. Since this model fills the space between polymer chains with a Newtonian fluid, it does not (in its present form) describe polymer melts. Effects of Chain Architecture This section treats considers the effect of chain architecture on a chain’s self-diffusion coefficient. The following analysis is based on the universal scaling law (eq 1) for D. The hydrodynamic scaling model is used to estimate a , u, and DO for chain and star polymers. A change in chain architecture is predicted to have a substantial effect on a but only a weak effect on u. How does chain architecture influence chain dimensions? Wheeler, Lodge, et ala’s on DO of linear polymers, 3-armed stars, and 12-armed stars is fit well by
Do = D M F E
(2)
DMf and g are given in Table I, separate values of DMf and g being required for different numbers ( f ) of arms. From Table I, D M ~ : D M ~ :=D1:1.17:1.33. M~~ g of a 2-arm star is slightly less than g of 3- and 12-arm stars, so, for a polymer having a nominal M = 1 X lo6 Da, DOof a linear chain is larger than Do of a 3-armed star but is less than DO of a 12-arm star. From eq 2 and Table I, for polymers with M = 1 X 106 the infinite-dilution hydrodynamic radii rh/ are in the ratio rh2:rh3:rhl2 = 1:1.04:0.91. Chain size can also be described with the radius of gyration. For an f-armed star polymer of arm molecular weight Ma, Grest and Kremer noteI5
-
1.18 0.41
(3) RE: M, f Since Ma = M / f , for chains of total molecular weight M
2744
Macromolecules, Vol. 23, No. 10, 1990
Phillies
-M
1.18 -0.77
f
(4) In a 0 solvent, Re2 MI. Lodge et a1.2 report Rg measurements from static light scattering by linear and star polymers. Their experimental results are compared with eq 4 in Table 11. Equation 2 was used to transform measured R, values to a single M , the same for all f. Agreement off dependence between experiment and theory is excellent. The universal scaling law eq 1 has previously been derivedgJOfrom a treatment of hydrodynamic interactions between two polymer chains, both chains being described a t the Kirkwood-Risemann bead-spring level. The detailed order of bonds between the beads plays no role in the calculation. Instead, each chain is treated as having a radially symmetric density g(s) of beads, s = 0 being the chain’s hydrodynamic center. A t this level of approximation, for a probe a in a matrix b R&r‘
-
C being a constant which is independent of the molecular weight of the polymer. Suppose in thisgJOderivation that a linear probe (a 2-armed star) were to be replaced by a star polymer (an f-armed star) having the same M. The number of polymer beads in the linear probe and the star probe would be equal. The list of bead-bead interactions between pairs of chains is independent off. There is one point in the derivation of eq 1 at which f is important, namely, the moment at which Rgaof eq 5 is evaluated for chains having the same M but different f. Noting that Rgb and C are both independent of probe f, while a concentrated system approximates 8 conditions af
-
P 5 f -0’38Rgb3/Mb
(6)
or 0.5 0 2
(7) M, f . N is implicitly dependent on c because Rg2 c - ~with x = I / 4 , l 6 so a c - ~ ~ In . terms of eq 1, u and x are related via u = 1 - 2x. It is possible (adequate data appear to be lacking) that u should be predicted to depend on f. Namely, as f is increased, the core of the star polymer becomes increasing dense and hence resistant to compression by neighboring polymer molecules. With increasing f, x might therefore tend to 0, in which case one would find u 1 as f 03. For fixed Ma, Lodge et a1.2 in fact note that u increases slightly (by perhaps 0.1) between f = 2 and f = 12. The above considerations lead to qualitative predictions as to the relative D, of linear and star polymers. The behavior of D f of f-armed stars is well-described by the parameter ‘yf
-
-
- -
introduced by Lodge et a1.2 To facilitate graphing the behavior of polymers of equal M , Lodge et al.’s data on 1.05 x 106 Da linear probes in a 1.3 X lo6 Da matrix are used as a reference system. For this reference system, a = 0.645 (with c in mg/mL) and u = 0.525. Equation 6 predicts, for a 1.05 X lo6 X Da 12-armed star probe in the same 1.3 X lo6 Da matrix, that a = 0.33. If u is the same for 2-armed and 12-armed stars, the qualitative prediction of the hydrodynamic scaling model is shown by Figure 1, line a. Over most of the accessible concentration range, K 1 2 . 2 gradually increases
Table I1 Theoreticalx6and Experimental2Radii of Gyration of f-Armed Stars, Reduced to Constant M, in Units of Rs2
f
(R&//Rg,Z)theo
1
2 3
0.85
12
0.50
(Rg,//R#.P)s.ptl
1 0.87 0.51
log ( c ) Figure 1. Hydrodynamic scaling predictions of K l z , ~=
(D12/
D~,lz)/D&,,z for 2- and 12-armed probes having equal total molecular weight (solid line) and equal arm molecular weight (dotted line). Dashed line (see text) shows how small parametric adjustments can yield excellent agreement between experi-
ment and eq 1.
from 1.0 to 2.0. For polymers of equal total molecular weight, the hydrodynamic scaling model thus predicts that linear chains are slightly more retarded than are 12-armed stars by a polymeric matrix. The details of the retardation are quite sensitive to the numerical parameters used to compute K12,2. Line b of Figure 1 shows how K12,2is changed if a12 and u are each slightly increased, namely, to their experimental values (0.39 and 0.608, respectively) for a 1.11 X lo6 Da 12-armed star in the matrix polymer solution of the previous paragraph. With this small change in parameters, K12,2at first gradually increases (to ~ 2 . 0 and ) then falls again. In contrast to behavior a t constant total molecular weight, one may consider behavior a t constant arm molecular weight Ma. The experimental reference system is taken to have 1.11 X lo6 Da 12-armed stars as probes in a 1.3 X lo6 amu matrix. For this reference system, experiment finds a = 0.390 and u = 0.608. From eq 7, the corresponding 2-armed star is predicted to have M = 2 X lo5 Da, u = 0.608, and a = 0.273. For 2- and 12-armed stars of equal M a , K12,2 is described by line c of Figure 1. At constant arm molecular weight, the hydrodynamic scaling model is seen to predict that a concentrated matrix polymer solution is much more effective-by an order of magnitude or more a t large c-at retarding star polymers than at retarding linear polymers. Lodge et al.1.2 give data on 3- as well as 12-armed stars. From ref 2, Table VIII, and applying R, W/2,R, of polymers having the same M differs by perhaps 13% between 2- and 3-armed stars, implying that a of 2- and 3-armed stars also differs by 13% or so. For stars having equal M and u, a t elevated c K3.2 would therefore be expected to be slightly larger than 1. For reasons not presently completely understood (though see below), u
-
Hydrodynamic Scaling Picture of Polymer Dynamics 2745
Macromolecules, Vol. 23, No. 10, 1990 Table 111 Values of a and v from Lodge et al.2 for Systems Having M = P
matrix M 1.3 X lo6 6.3 x 105 1.4 x 105
probe M f=2 1.05 X lo6 4.22 x 105 1.79 x 105
0.645 0.312 0.140
0.525 0.589 0.663
1.19 x 106 3.79 x 105 3.79 x 105
f=3 1.3 X lo6 6.3 x 105 1.4 x 105
0.657 0.179 0.139
0.657 0.681 0.679
1.3 X 1.3 X 1.4 x 1.4 X
f=12 1.69 X lo6 1.11 x 106 4.67 x 105 5.5 x 104
0.413 0.390 0.164 0.0638
0.617 0.608 0.666 0.757
lo6 lo6 105 los
a
Y
m Figure 2. Dependence of a on ( M P ) 1 / 2for 2-, 3-, and 12-armed stars. Solid line is a best-fit. Dashed and dotted lines had their vertical shifts determined by eq 6.
Table IV Fits of a (Table 111) to a = a ~ M and g Root-Mean-Square Errors (Rmse) in These Fits
f 2 3 3
12 12
aM
1.59 x 1.05 x 3.87 x 1.11 x 7.67 x
10-5 10-5 10-5 10-5 10-7
g
% rmse
0.756 0.756 0.657 0.756 0.96
6 17 11 29 26
differs slightly between different probe:matrix systems. The consequences of a 13% change in a between f = 2 and f = 3 can be swamped by changes in u between linear polymers and simple stars. The effect on K12,2 of the large change between f = 2 and f = 12 would be much more difficult to mask by changing Y, so the above discussion has emphasized K12,~. The hydrodynamic scaling model directly predicts a, not K I , ~the , direct test of the hydrodynamic scaling model being compared to eq 6 with af. The self-similarity approximation implicitly assumes that the probe and matrix polymers are largely identical. This is a reasonable approximation if the matrix and probe molecular weights M and P are equal, but the approximation presumably deteriorates if M > P. Examination of measured a’s is limited to those of Lodge et al.’s systems2 with M P, as listed in Table 111. Table IV gives results from fits (at fixed f ) of these data to the form a=a ,@
(9)
For 2-armed stars g = 0.756 in the best fit. For 3- and 12-armed stars, Table IV includes fits using both aM and g as free parameters and also fits to aM made while holding g a t its best value for f = 2. Figure 2 plots a from Table I11 against (MP)llZ. The solid line represents the best fit to a for systems with f = 2. The dashed and dotted lines (which refer to f = 3 and 12, respectively) are drawn parallel to the solid line, using eq 6 to fix the ratio of the y intercepts. The hydrodynamic scaling model correctly predicts that a3 and a 1 2 are both less than az, a3 being somewhat scattered, while experimentally a12 is typically slightly larger than predicted. Discussion As shown above, the hydrodynamic scaling model is successful at reproducing the major qualitative features of chain and star diffusion through matrix polymer solutions. Quantitative agreement with the simplified calculation of ref 9 is not perfect. Comparing probes of equal total molecular weight, the model correctly predicts that
a concentrated matrix polymer is more somewhat effective a t retarding a linear chain that a retarding a manyarmed star. Comparing probes of equal arm molecular weight, the model correctly predicts that a concentrated matrix polymer is substantially more effective at retarding a many-armed star than a t retarding a linear chain. Experimental measurements of D1 and D* thus do not, at the present stage of experimental and theoretical development, serve to distinguish between hydrodynamic and entanglement-based pictures of polymer dynamics. In refs 1 and 2, several other criticisms of the hydrodynamic scaling model were advanced. In particular, in Wheeler and Lodge’s fits of eq 1 to their data, DO was alternatively used as a fitting parameter and as the c 0 limit of D(c). The c 0 limit of DOwas consistently smaller than the corresponding DO from a three-parameter fit to D(c), especially for large-M probes in large-M matrices. The difference between fitted and measured DO has been attributed1’ to probe coil contraction (which increases D) with increasing c. Wheeler and Lodge’ argue that the difference in the DO’Sis not due to coil contraction because “the derivation of the stretched exponential advanced by Phillies already incorporates the probe coil contraction, and thus it should not be incorporated twice”. A careful examination of the derivation of the universal scaling equationlo shows, contrary to Wheeler and Lodge, that self-consistency requires including the consequences of probe coil contraction both in DOand in the exponential. A brief sketch of the universal scaling law’s derivation should clarify this consistency requirement. The extant derivation of the universal scaling law treats the drag coefficient f(c,a) of a polymer coil of hydrodynamic radius a immersed in a polymer solution having concentration c. f(c,a)and the self-diffusion coefficient are linked by the Einstein diffusion equation D, = kBT/ f(c,a). For a small concentration increment 6c
-
-
Since a depends but weakly on c, the final term is negligible by comparison with its neighbors. Equation 10 is integrated by imposing a self-similarity assumption (written here for the case that the matrix and probe polymer are the same species)
Physically, eq 11 implies that the ability of an additional matrix polymer to retard the motion of the probe coil is proportional to the matrix polymer’s hydrody-
2746
Phillies
Macromolecules, Vol. 23, No. 10, 1990 on the probe chain is best stated in terms of the polymer probe's mobility p = l/f. For polymer chains whose hydrodynamic centers are separated by R, a series expansion in a / R (where R = IRI) givedo to lowest nonzero order
I being the identity matrix and denoting the unit vector. For one probe in a solution containing N other chains
(I-E N
v==o
,
9a4fkjRj
=
I
F
(14)
)
R, being the vector separating the hydrodynamic cenI
0.5
I
I
1.0
1.5
ters of the probe and matrix polymer j . In this equation, p is implicitly concentration-dependent because the polymer hydrodynamic radius a depends on c, namely, a2 Mc-x (except at very small c, where x = 0). Thus, a4 c - ~ ~Furthermore, . PO refers to the actual hydrodynamic radius of the first polymer chain, not to the radius of the probe chain in the absence of the matrix chains, so po is weakly concentration-dependent through its inverse proportionality to a. In other words, po is the mobility (and D = p o k ~ Tis the diffusion coefficient) which the polymer chain would have at infinite dilution, if a t infinite dilution the chain had the same hydrodynamic radius which the chain has at the concentration c. Polymers in solution do not occupy fixed relative positions, so eq 13 is properly averaged over the R,. Comparing eqs 10,11, and 13
--
log c (q/lOO ml)
K ( a ) = Ca4 (15) C being a concentration-independent constant embodying the hydrodynamic series expansions and ensemble averages described in ref 10. It follows that m(c,a) = p,, exp(-JcCa4 dc/M)
I
55
I
6.0
I
65
log M
Figure 3. A. Zero-shear viscosity against concentration for polystyrene: toluene at 40 "C, based on measurements of Onogi et al.,13 as presented by Doi and Edwards.14 Solid lines: fits
to the stretched exponential of eq 18, lines representing surfaces of constant M. B. Zero-shear viscosity against polymer molecular weight for polystyrene: toluene, 40 "C, based on Onogi et al.,13from Doi and Edwards.14 Solid lines: fits to the stretched exponential of eq 18, lines representing surfaces of constant c. namic interaction with the solvent, Le., to the matrix polymer's drag coefficient f at concentration c. The integral of eq 10, under the approximation daldc = 0, is f(c,a) = f oe x p ( L c Kdc)
(12)
To apply eq 12, a form is needed for K. K is assumed to be dominated by the hydrodynamic interactions between pairs of nearby polymer chains. The effects of threebody chain-chain-chain interactions are approximated by the self-similarity assumption (eq 11);three-body interactions contribute to the difference between f(c,a) and f(0,a).
For an isolated polymer coil in solvent, f o = 6xqa, a being the hydrodynamic radius of the polymer and 7 being the solvent viscosity. The effect of a second polymer chain
(16)
M is the polymer molecular weight, appearing here because the summation variable j has been replaced by the mass concentration c. In eq 16, po refers to the probe chain as it behaves in the presence of the other chains. PO is is weakly concentrationstill proportional to a-I, so dependent. Thus, in applying eq 16 or its diffusional equivalent D
kBT~(c,a) = Do exp(-fCa4 dc/M)
(17)
DOand a should both be interpreted to be c-dependent. Wheeler and Lodge1v2further argue that the hydrodynamic scaling model has not received universal acceptance because (among other reasons) the model does not account for all features of the viscoelastic properties of concentrated solutions, such as the plateau modulus. The model's prediction of stretched-exponential c and M dependences does, however, describe measurements of the zeroshear viscosity 7 of random-coil polymers. For example, parts A and B of Figure 3 show data of Onogi et al.13 on q of po1ystyrene:toluene at 40 "C, as presented by Doi and E d ~ a r d s . 1Doi ~ and Edwards14 describe Figure 3B as confirming 7 M x with x == 3.4, the result 7 M x with x = 3.0 being a major prediction of the reptationscaling model. The data in parts A and B of Figure 3 are in fact well-described by stretched exponentials, solid lines in these figures representing
-
-
Macromolecules, Vol. 23, No. 10, 1990
Hydrodynamic Scaling Picture of Polymer Dynamics 2747
Table V Nonlinear Least-Squares Fits of Stretched Exponentials of 7 of Polystyrene/Toluene (40 “C) Using Data of Onogi et al.*aAs Given by Doi and Edwards“
D = Do exp[-acu]
Part I
M , kDa 206 357 557 656 756 923 1050 1370 1670
% rmse
Y
0.634 1.13 1.34 1.44 1.65 1.77 1.78 2.06 2.29
0.676 0.603 0.594 0.606 0.580 0.583 0.606 0.589 0.578
11
13 10 6.3 16 17 12 10 13
Part I1 c, g/100 mL
4
0.932 0.814 0.679 0.586 0.517 0.493
6
9.7 14 18 20
Y
70 rmse
0.502 0.467 0.44 0.425 0.388 0.392
7.1 10 7.5 13 11 13
Part 111 Figure 3A Figure 3B
ffM
V
Y
% rmse
1.79 1.66
0.590 0.597
0.426 0.415
18 18
Part I of this table uses eq 19 to fit points from Figure 3A at fixed M part I1 of this table uses eq 20 t o fit points from Figure 3B at fixed c; part I11 of the table describes fits of eq 18 to all the data in parts A and B of Figure 3. a,Y, and y are for units: c in g/100 mL,M = 1 for a 100 000 Da polymer.
where qo = 0.471 CPis the solvent viscosity. For all the data in Figure 3A, a nonlinear least-squares fit to eq 18 with a , u, and y as free parameters obtains a = 1.788, u = 0.590, and y = 0.426. For all the data in Figure 3B, the same process obtains a = 1.664, u = 0.597, and y = 0.415. The root-mean-square error between either fit and its data set was approximately 18%. Parts A and B of Figure 3 represent largely overlapping sets of measurements, so the differences between a,u, y from these two parts are a robust estimator of the experimental error in determining the parameters. Hence, u and y are obtained with an error of f l % or so, while a was obtained with an error of f 4 % . As noted elsewhere’s it is easier to determine u than to determine a. Table V shows results from nonlinear fits to obtain a, u, and y for individual values of M (Figure 3A) or c (Figure 3B). If data from Figure 3A (which gives q against c ) for fixed M are fit to = vo exp(-a,c’) (19) one finds that u falls with increasing M, ranging from u = 0.675 a t M = 2 X los to u = 0.578 at M = 1.67 X lo6. If data from Figure 3B (which gives q against M) for fixed c are fit to 7
= qo exp(-aMMY) (20) one finds that y falls with increasing c, ranging from y = 0.502 a t c = 4% to y = 0.388 at c = 18%. The analytic form (eq 18) predicted by the hydrodynamic scaling model thus does give a good account of the dependence of q on polymer concentration and molecular weight. Equation 18 (with constant exponents) becomes increasingly reliable a t large M and c. Wheeler and Lodge’ advance the argument that “the stretched exponential is rather sensitive to changes in CY
- I .2 I
45
I
5.0
,
6
1
1
,
,
,
,
l
5.5
,
,
,
6.0
,
6.5
log,o (MP)‘’2
-
Figure 4. a against M for all of Wheeler and Lodge’s data using linear probes, showing good agreement with a (MP)’/*.
... the uncertainty in the experimental data is generally much too small to generate noticeable error bars on values of a ... Thus the wide scatter seen (in ref 2, Figure 17) in values of a ... must arise from inadequacies in the
theory ...”. First, in their Figure 17, Wheeler and Lodge include results on 2-, 3-, and 12-armed stars. Even if
a = k(MP)”* (21) for each type of star separately, there is no reason to suppose that k will be the same for f = 2,3, and 12. Indeed (cf. Figure 2), the hydrodynamic scaling model predicts that k depends on f. An appropriate test of eq 21 involves only one type of star at a time, not the three different types included in ref 2, Figure 17. Figure 4 shows an appropriate comparison of CY measurements, plotting Lodge and Wheeler’s a against M for PS with a single f , viz. f = 2. Most points are within 10% of a line slope 1. The scatter in values of a is not large, just as expected if the hydrodynamic scaling theory and its prediction a (MP)II2were basically adequate to treat diffusion. Second, with respect to their argument that a should not be scattered, note Table V.iii. Between parts A and B of Figure 3, there is a 7% change in a,even though the two parts include many of the same data points. The random noise in determining a,even with good data, is large. Finally, Wheeler and Lodge1V2 argue that their comparison of u and M (ref 2, Figure 18) does not show the predicted transition from the small-M, u = 1limit to the large-M, u = ‘ / z limit. Their figure tends to obscure the transition apparent in their data. Namely, within the hydrodynamic scaling model, this predicted transition is driven by the change from small-M to large-M chain contraction. While large linear chains and large stars all contract with increasing matrix concentration, for fixed polymer molecular weight the exact degree of contraction (as measured by x of the scaling law R, c x ) is expected to be sensitive to the number of arms and to the relative molecular weights of probe and matrix polymers. For each f and MIP, u should show a clean dependence on polymer size. The transition need not occur a t the same P for each f . Indeed, as noted in ref 2, u is larger for stars than for linear polymers, as would occur of the transition required a larger M for star polymers than for linear polymers. Combining u values for different f and MIP, as done in ref 2, Figure 18, therefore tends to mask the transition. It is not here claimed that the hydrodynamic scaling picture is in all respects complete. As presently developed, the model does not include detailed consequences of chain rigidity, short-range hydrodynamic interactions, polymer internal modes, or the detailed form of
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the chain-chain correlation hole. Nor has the model yet been extended to treat other solution transport properties, such as probe diffusion, sedimentation, or the piateau modulus. A rigorous justification of the self-similarity approximation-one capable of predicting correction terms in a systematic way-is presently lacking.
References and Notes Wheeler, L. M.; Lodge, T. P. Macromolecules 1989,22, 3399. Lodge, T. P.; Markland, P.; Wheeler, L. M. Macromolecules 1989,22,3409. de Gennes, P.-G. Scaling Concepts in Polymer Physics; Corne11 University Press: Ithaca, NY, 1979. Hess, W. Macromolecules 1986, 19, 1395. Hess, W. Macromolecules 1987,20, 2587. Skolnick, J.; Yaris, Y.; Kolinski, A. J . Chem. Phys. 1987, 88, 1407. Skolnick, J.; Yaris, Y. J . Chem. Phys. 1987, 88, 1418.
(8) Phillies, G. D. J. Macromolecules 1986, 19, 2367. (9) Phillies, G. D. J. Macromolecules 1987, 20, 558. (10) Phillies, G. D. J. Macromolecules 1988,21, 3101. (11) Phillies, G. D. J. J . PhYs. Chem. 1989,93,5029. (12) Fujita, H.; Einaga, Y. P o h . J . (Tokyo) 1985, 17, 1131. (13) Onogi, S.; Kimura, S.; Kato, T.;Masuda, T.;Miyanaga, N. cited in ref 14 as J . Polym. Sci., Part C 1966, 381. (14) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (15) Grest, G. S.; Kremer, K. Springer Proceedings in Physics.
Computer Simulation Studies in Compressed Matter Physics; Springer-Verlag: Berlin, 1988; Vol. 33, p 76. (16) Daoud, M.; Cotton, J. P.; Farnoux, B.; Jannink, G.; Sarma, G.; Benoit, H.; Duplessix, R.; Picot, C.; de Gennes, P.-G. Macromolecules 1975, 8, 804. (17) Wheeler, L. M.; Lodge, T. P.; Hanley, B.; Tirrell, M. macromolecules 1987, 20, 1120. (18) Phillies, G. D. J.; Gong, J.; Li, L.; Rau, A,; Zhang, K.; Rollings, J.; Yu, L-P., J . Phys. Chem. 1989, 93, 6219.
Enantioselective Permeation of a-Amino Acid Isomers through Poly(amino acid)-Derived Membranes Atsushi Maruyama,' Noriyuki Adachi, Takehisa Takatsuki, Masanori Torii, Kohei Sanui, and Naoya Ogata Department of Chemistry, Faculty of Science and Technology, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102, Japan Received September 7, 1989;Revised Manuscript Received December 1, 1989
ABSTRACT Poly(L-glutamates) having amphiphilic side chains of (n-nonylphenoxy)oligo(oxyethylene) (NONx-PLG) were observed to possess potential properties as membrane materials for resolving optical isomers of a-amino acids. Permeation rate ratios of >8.0 for the D isomer to L isomer of tyrosine and tryptophan were achieved. Permeation of amino acids for times >500 h did not cause any drop in selectivity. Operating temperature considerablyaffected the permselectivityof the isomers. Since NONr-PLG adopted mainly an a-helical structure in its cast film and formed a liquid crystalline (nematic) type mesophase,this ordered structure of poly(amino acid) derivatives was considered to be the reason for the enantioselectivity of the membrane.
Introduction With the rapid production of novel specialty (highly functional) materials such as pharmaceutics, bioactive substances, and liquid crystals, improvement of preparative separation methods for geometrical isomers including optical isomers is currently demanded. Although the membrane separation process has marked advantages in production such as ease of handling, instrumental simplicity, a n d efficiency in energy as compared t o t h e chromatographic separation process, a practical membrane having the capability of separating optical isomers has not yet appeared except in the form of liquid membranes. These are triphasic systems composed of aqueous-rganic-aqueous solutions utilizing optically active crown ethers, originally demonstrated by Cram et al.' The permeation of substances through a membrane is determined by the partitioning behavior of the substances with respect to the membrane and their diffusibility (or mobility) in the membrane. Thus, to improve enantioselectivity of a membrane, considerable effort should be made to optimize these two factors. The par-
* To whom correspondence should be addressed. 0024-9297/90/2223-2748$02.50/0
titioning behavior of optical isomers is largely influenced by the structure and number of recognition sites in the membrane. However, diffusibility is difficult to control because optical isomers have identical molecular sizes. To refine the selectivity of polymeric membranes, we paid particular attention to the microenvironmental nature in the vicinity of the recognition sites as well as the distribution (or assembling) mode of the recognition sites. Aqueous liquid membranes with amphiphilic substances such as surfactants were reported to discriminate geometric isomers of amphiphilic substances or of hydrocarbons in terms of their permeation or partitioning behavior based on the differential solubilizing effect.2 In this case, the hydrophilic-hydrophobic balance of amphiphilic substances was considered to play a significant, role. Colonna et al.3 reported the successful asymmetric oxidation of electron-poor olefins such as chalcones to optically active oxiranes in the triphase system (water/organic solvent-poly(amino acid)). In this case, the poly(amino acid), which provides the stereoselectivity of the reaction, was placed at the interface of an emulsion of the organic phase in the aqueous phase. The hydro0 1990 American Chemical Society
